NMR Relaxation

1
NMR Relaxation

• After an RF pulse system needs to relax back to
  equilibrium condition
• Related to molecular dynamics of system
• may take seconds to minutes to fully recovery
• usually occurs exponentially
• (n-ne)t displacement from equilibrium value ne at
  time t
• (n-ne)0 at time zero
• Relaxation can be characterized by a time T
• relaxation rate (R) 1/T
• No spontaneous reemission of photons to relax
  down to ground state
• probability too low ? cube of the frequency
• Two types of NMR relaxation processes
• spin-lattice or longitudinal relaxation (T1)
• spin-spin or transverse relaxation (T2)

z
z
z
Mo
Mo
B1 off (or off-resonance)
T1 T2 relaxation
x
x
x
B1
Mxy
w1
y
y
y
w1
2
NMR Relaxation

• Spin-lattices or longitudinal relaxation
• Relaxation process occurs along z-axis
• transfer of energy to the lattice or solvent
  material
• coupling of nuclei magnetic field with magnetic
  fields created by the ensemble of vibrational and
  rotational motion of the lattice or solvent.
• results in a minimal temperature increase in
  sample
• Relaxation time (T1) ? exponential decay

Mz M0(1-exp(-t/T1))
3
NMR Relaxation

• Spin-lattices or longitudinal relaxation
• Relaxation process occurs along z-axis
• Measure T1 using inversion recovery experiment

4
NMR Relaxation

• Spin-lattices or longitudinal relaxation
• Collect a series of 1D NMR spectra by varying t
• Measure T1 using inversion recovery experiment

5
1
2
6
NMR Relaxation

• Spin-lattices or longitudinal relaxation
• Collect a series of 1D NMR spectra by varying t
• Plot the peak intensities as a function of t fit
  to an exponential

7
NMR Relaxation

• Mechanism for Spin-lattices or longitudinal
  relaxation
• Dipolar coupling between nuclei and solvent (T1)
• interaction between nuclear magnetic dipoles
• depends on correlation time
• oscillating magnetic field due to Brownian
  motion
• depends on orientation of the whole molecule
• in solution, rapid motion averages the dipolar
  interaction Brownian motion
• in crystals, positions are fixed for single
  molecule, but vary between molecules
• leading to range of frequencies and broad
  lines.

Tumbling of Molecule Creates local Oscillating
Magnetic field
8
NMR Relaxation

• Mechanism for Spin-lattices or longitudinal
  relaxation
• Solvent creates an ensemble of fluctuating
  magnetic fields
• causes random precession of nuclei ? dephasing
  of spins
• possibility of energy transfer ? matching
  frequency

Field Intensity at any frequency

• tc represents the maximum frequency
• 10-11s 1011 rad s-1 15920 MHz
• All lower frequencies are observed

9
NMR Relaxation

• Mechanism for Spin-lattices or longitudinal
  relaxation
• Intensity of fluctuations in magnetic fields due
  to Brownian motion as a function of frequency
• causes random precession of nuclei ? dephasing
  of spins
• possibility of energy transfer ? matching
  frequency

tc 10-8 s-1
Spectral Density Function (J(w))
tc 10-9 s-1
Increasing MW
tc 10-10 s-1
tc 10-11 s-1
10
NMR Relaxation

• Spin-lattices or longitudinal relaxation
• Relaxation process in the x,y plane
• Related to peak line-width
• Inhomogeneity of magnet also contributes to peak
  width
• T2 may be equal to T1, or differ by orders of
  magnitude
• T2 can not be longer than T1
• No energy change

T2 relaxation
(derived from Heisenberg uncertainty principal)
11
NMR Relaxation

• Spin-spin or Transverse relaxation
• exchange of energy between excited nucleus and
  low energy state nucleus
• randomization of spins or magnetic moment in
  x,y-plane
• related to NMR peak line-width
• relaxation time (T2)

Mx My M0 exp(-t/T2)
Please Note Line shape is also affected by the
magnetic fields homogeneity
12
NMR Relaxation

• Spin-spin or Transverse relaxation
• While peak width is related to T2, not an
  accurate way to measure T2
• Use the Carr-Purcell-Meiboom-Gill (CPMG)
  experiment to measure spin-echo
• Refocuses spin diffusions due to magnetic field
  inhomogeneiety

13
NMR Relaxation

• Spin-spin or Transverse relaxation
• Collect a series of 1D NMR spectra by varying t
• Plot the peak intensities as a function of t and
  fir to an exponential
• Peaks need to be resolved to determine
  independent T2 values

Mx My M0 exp(-t/T2)
Biochemistry 1981, 20, 3756-3764
14
NMR Relaxation

• Mechanism for Spin-lattices and Spin-Spin
  relaxation
• Relaxation is related to correlation time (tc)
• Intensity of fluctuations in magnetic fields due
  to Brownian motion as a function of frequency
• MW ? radius ? tc

where r radius k Boltzman constant h
viscosity coefficient

• rotational correlation time (tc) is the time it
  takes a
• molecule to rotate one radian (360o/2p).
• the larger the molecule the slower it moves
• T2 T1
• small molecules (fast tc) T2 T1
• Large molecules (slow tc) T2 lt T1

15
NMR Relaxation

• Mechanism for Spin-lattices and Spin-Spin
  relaxation
• Illustration of the Relationship Between MW, tc
  and T2

16
NMR Relaxation

• Mechanism for Spin-lattices and Spin-Spin
  relaxation
• Relaxation is related to correlation time (tc)
• intramolecular dipole-dipole relaxation rate of
  a nuclei being relaxed by n nuclei

Depends on nuclei type
Extreme narrowing limit
Depends on distance (bond length)
17
NMR Relaxation

• Mechanism for Spin-lattices and Spin-Spin
  relaxation
• Relaxation is related magnetic field strength (w)

T1 minima and values increase with increasing
field strength
T2 reduced at higher field strength for larger
molecules leading to broadening
18
NMR Relaxation

• Mechanism for Spin-lattices and Spin-Spin
  relaxation
• Different relaxation times (pathways) for
  different nuclei interactions
• 1H-1H ? 1H-13C ? 13C-13C
• relaxation rates depend on the number of
  attached nuclei and bond length
• carbon 13C gt 13CH gt 13CH2 gt 13CH3
• proton dominated by relaxation with other
  protons in molecule
• Same general trends as intramolecular relaxation

Extreme narrowing limit
19
NMR Relaxation

• Typical Spin-lattices Relaxation Times
• T2 T1
• Examples of 13C T1 values
• number of attached protons greatly affects T1
  value
• Non-proton bearing carbons have very long T1
  values
• T1 longer for smaller molecules
• Differences in T1 values related to local motion
• Faster motion ? longer T1
• Solvent can affect T1 values

Solvent Effects
CH3OH
CD3OD
20
NMR Relaxation

• Chemical Shift Anisotropy Relaxation
• Remember

• Magnetic shielding (s) depends on orientation of
  molecule relative to Bo
• magnitude of s varies with orientation

Bo
Solid NMR Spectra
Orientation effect described by the screening
tensor s11, s22, s33 If axially
symmetric s11 s22 s s33 s-
21
NMR Relaxation

• Chemical Shift Anisotropy (CSA) Relaxation
• Effective Fluctuation in Magnetic field strength
  at the nucleus
• Causes relaxation
• not very efficient
• in extreme narrowing region
• depends strongly on field strength and
  correlation time
• depends strongly on chemical shift ranges
• results in line-broadening
• increase in sensitivity and resolution at higher
  field strengths may be overwhelmed by CSA affects

22
NMR Relaxation

• Chemical Shift Anisotropy (CSA) Relaxation
• Line-shape increases as CSA increases with
  magnetic field strength

Two peaks in nitrogen doublet experience
different CSA contributions
Can improve line shape if only select this peak
Nature Structural Biology  5, 517 - 522 (1998)
23
NMR Relaxation

• Chemical Shift Anisotropy (CSA) Relaxation
• Line-shape increases as CSA increases with
  magnetic field strength

Peaks originating from 195Pt-1H2 coupling are
broadened at higher field due to CSA (shortening
of T1(Pt)
Increasing Magnetic Field
24
NMR Relaxation

• Quadrupolar Relaxation
• Quadrupole nuclei (I gt ½)
• Introduces a second and very efficient
  relaxation mechanism
• a factor of 108 as efficient of dipole-dipole
  relaxation
• Distribution of charge is non-spherical
  ?ellipsoidal
• for I ½, charge is spherically distributed

25
NMR Relaxation

• Quadrupolar Relaxation
• Electric Field Gradient (EFG)
• tensor quantity
• can be reduced to diagonal values Vxx,Vyy,Vzz
• Vxx Vyy Vzz 0
• asymmetry factor (h)
• Vxx,Vyy,Vzz are calculated from the sum of
  contributions from all charges qi at a distance
  ri

26
NMR Relaxation

• Quadrupolar Relaxation
• Factors affecting quadrupolar relaxation
• Depends strongly on nuclear properties
• quadrupole moment (Q) and spin number (I)
• Depends strongly on molecular properties
• correlation time (tc)
• increasing temperature increases tc and
  increases relaxation time and reduces resonance
  linewidth
• shape (Vzz, h)
• Depends primarily on electric field gradient
  (EFG)
• can vary from zero to very large numbers
• charge close to nucleus have predominating
  effect (distance dependence)
• movement of molecules in liquid reduces distance
  effect to zero
• solids with fixed distances have contributions
  from distant charges

27
NMR Relaxation

• Dipole nuclei (I1/2) coupled to quadrupole
  nuclei (Igt1/2)
• Quadrupole relaxation significantly broadens
  nuclei
• obscures spin-splitting pattern
• If quadrupole relaxation is slow, broadening is
  diminished and spin-splitting pattern is observed

Very short T1 average value
Increasing T1
Increasing T1
Long T1 normal splitting
28
NMR Relaxation

• Dipole nuclei (I1/2) coupled to quadrupole
  nuclei (Igt1/2)
• Quadrupole relaxation significantly broadens
  nuclei through scaler coupling
• Lowering temperature can sharpen peaks broaden
  by quadrupole relaxation
• lower temperature ? increase tc ? shorten T1Q

29
NMR Relaxation

• Quadrupolar Relaxation
• If the system is axially symmetric, h 0 and Vxx
  Vyy
• Only need to determine Vzz
• equal distribution of three charges around the
  z-axis at a distance r from N

30
NMR Dynamics and Exchange
Despite the Typical Graphical Display of
Molecular Structures, Molecules are Highly
Flexible and Undergo Multiple Modes Of Motion
Over a Range of Time-Frames
DSMM - Database of Simulated Molecular
Motions http//projects.villa-bosch.de/dbase/dsmm/

Click on image to start dynamics simulation
31
NMR Dynamics and Exchange

• Multiple Signals for Slow Exchange Between
  Conformational States
• Two or more chemical shifts associated with a
  single atom/nucleus

Populations relative stability
Rex lt w (A) - w (B)
Exchange Rate (NMR time-scale)

• Factors Affecting Exchange
• Addition of a ligand
• Temperature
• Solvent

32
NMR Dynamics and Exchange
Different environments
OH exchanges between different molecules and
environments. Observed chemical shifts and
line-shapes results from the average of the
different environments.
Intermediate exchange Broad peaks
Slow exchange CH2-OH coupling is observed
Fast exchange Addition of acid CH2-OH coupling
is absent
33
NMR Dynamics and Exchange
Effects of Exchange Rates on NMR data
k p Dno2 /2(W1/2)e (W1/2)o)
k p Dno / 21/2
k p (Dno2 -  Dne2)1/2/21/2
k p ((W1/2)e-(W1/2)o)

• k exchange rate
• W1/2 peak-width at half-height
• peak frequency
• e with exchange
• o no exchange

Dno
34
NMR Dynamics and Exchange
Equal Population of Exchange Sites
40 Hz
No exchange
k 0.1 s-1
slow
k 5 s-1
k 10 s-1
With exchange
k 20 s-1
k 40 s-1
Increasing Exchange Rate
coalescence
k 88.8 s-1
k 200 s-1
k 400 s-1
k 800 s-1
k 10,000 s-1
fast
35
NMR Dynamics and Exchange

• Example of NMR Measurement of Chemical Exchange
• Two different cyclopentadienyl groups in
  Ti(h1-C5H5)2(h5-C5H5)2
• Exchange rate changes as a function of
  temperature
• But, chemical shifts also change as a function
  of temperature

36
NMR Dynamics and Exchange

• Example of NMR Measurement of Chemical Exchange
• Multiple resonances may be affected by exchange
• Rotation about N-C bond
• different coalescence rates because of different
  na-nb

C3 C4 separation smaller than C6 C2
37
NMR Dynamics and Exchange

• Exchanges Rates and NMR Time Scale
• NMR time scale refers to the chemical shift time
  scale
• remember frequency units are in Hz (sec-1) ?
  time scale
• exchange rate (k)
• differences in chemical shifts between species in
  exchange indicate the exchange rate.

Time Scale Chem. Shift (d) Coupling Const.
(J) T2 relaxation Slow k ltlt dA- dB
k ltlt JA- JB k ltlt 1/ T2,A- 1/
T2,B Intermediate k dA - dB k
JA- JB k 1/ T2,A- 1/ T2,B Fast
k gtgt dA - dB k gtgt JA- JB k gtgt
1/ T2,A- 1/ T2,B Range (Sec-1) 0 1000 0 12
1 - 20
2
1
1
Slow exchange at -60o
38
NMR Dynamics and Exchange

• Exchange Rates and NMR Time Scale
• NMR time scale refers to the chemical shift time
  scale
• For systems in fast exchange, the observed
  chemical shift is the average of the individual
  species chemical shifts.

dobs f1d1 f2d2 f1 f2 1
where f1, f2 mole fraction of each
species d1,d2 chemical shift of each species
Fast exchange, average of three slow exchange
peaks
d 1.86 ppm 0.25 x 2.00 ppm
0.25 x 1.95 ppm 0.5 x
1.75 ppm
39
NMR Dynamics and Exchange

• Unequal Population of Exchange Sites
• differential broadening below coalescence
• lower populated peak broadens more

40 Hz
3
k 0.1 s-1
1
slow
k 5 s-1
Exchange rate depends on population (p)
k 10 s-1
k 20 s-1
k 40 s-1
Increasing Exchange Rate
coalescence
Above coalescence
k 88.8 s-1
k 200 s-1
k 400 s-1
k 800 s-1
fast
k 10,000 s-1
Weighted average
40
NMR Dynamics and Exchange

• Example of NMR Measurement of Chemical Exchange
• Unequal populated exchange sites
• exchange between axial and equatorial position
• exchange rate can be measured easily up to
  -44oC. Can easily measure na-ne and peak ratios
• again, different broadening is related to
  chemical shift differences between axial and
  equatorial positions
• difficult to determine accurate na-ne
• difficult to determine accurate k

41
NMR Dynamics and Exchange

• Use of magnetization transfer to study exchange
• Lineshape analysis is related to the rate of
  leaving each site
• no information on the destination
• problem for multisite exchange
• Saturation Transfer Difference (STD) Experiment
• Collect two spectra
• one peak is saturated (decoupler pulse)
• decoupler or saturation pulse is set far from
  any peaks (reference spectrum)
• subtract two spectra
• If nuclei are exchanging during the saturation
  pulse, additional NMR peaks will exhibit a
  decrease in intensity due to the saturation pulse.

A
B
Mz(0)
Mz(0)
decouple site A
MzA 0
Mz(0)
exchange from A to B
A
B
Exchange rate
T1 relaxation
42
NMR Dynamics and Exchange

• Use of magnetization transfer to study exchange
• at equilibrium (t8)
• kB can be measured from MZB(8), MZB(0) and T1B
• assumes T1A T1B
• if T1A ? T1B,difficult to measure T1A and T1B?
  partial average

or
43
NMR Dynamics and Exchange

• Use of magnetization transfer to study exchange
• p t p/2 pulse sequence
• exchange takes place during (t)

Saturate peak 2, exchange to peak 3
Saturate peak 1, exchange to peak 4
44
NMR Dynamics and Exchange

• Use of magnetization transfer to study exchange
• p t p/2 pulse sequence
• exchange takes place during (t)

Selective 180o pulse
Saturation transferred during t
Fit peak intensities to determine average T1 and
k (k15.7 s-1 T1 0.835 s)
45
NMR Dynamics and Exchange

• Activation Energies from NMR data
• rate constant is related to exchange rate
  (k1/tex)

Different nuclei and magnetic field strengths
Measure rate constants at different temperatures

• Calculating DH and DS may not be reliable
• temperature dependent chemical shifts
• mis-estimates of line-widths in absence of
  exchange
• poor temperature calibration
• signal broadened by unresolved coupling
• To obtain reliable DH and DS values
• obtain data over a wide range of temperature
  where coalescence points can be monitored
• measure at different spectrometer frequencies
• use different nuclei with different chemical
  shifts
• use line-shape analysis software
• use magnetization transfer

46
NMR Dynamics and Exchange

• Two-Dimensional Exchange Experiments
• Uses the NOESY pulse sequence (EXSY)
• uses a short mixing time ( 0.05s)
• exchange of magnetization occurs during mixing
  time
• NOEs will also be present
• need to distinguish between NOE and exchange
  peaks
• usually opposite sign

x
l